Table 1.
Comparisons to existing methods.
Fig 1.
a) An isolated cell is modeled as a disk. b) A cell is modeled as a disk segment when contacting other cell(s). An outer edge ei is an arc or a circle, representing the boundary between cell ci and the outside medium (denoted as c0). An inner edge ei,j occurs when a cell ci is in contact with another cell cj. Their shared boundary is modeled as a straight line segment. When two cells ci and cj make contact, their outer edges (arcs) ei and ej intersect at two vertices vi,0,j and vj,0,i, which are also the two end-points of the inner edge ei,j. c) When three cells ci,cj and ck intersect, they form a vertex vi,j,k. d) A cell completely surrounded by other cells is represented as a polygon.
Fig 2.
The boundary of a cell is formed by connected edges. It is modeled as an oriented closed curve in the counterclockwise direction. Each physical inner edge is represented twice using two half-edges, once each in opposite directions for each of the two contacting cells. Each physical outer edge is also represented twice with two half-edges, once for the cell, and once for the outside space.
Fig 3.
Forces applied to the junction vertex of three cells a, b, and c. The tension force T(ei,j) exerts along the direction ei,j of an inner edge (interior cell boundary), or along the tangent direction of outer edge ei (free cell boundary), where (i,j) are the two indices of cells a,b, or c. The pressure force P(ei,j) acts along the direction normal to the cell boundary, in the direction from the cell with higher pressure to that with lower pressure.
Fig 4.
Diagram representing the forces involved during cell growth. Vertices v1 and v2 are moved by Δv1 and Δv2 due to the forces Fv1 and Fv2, during the growth process respectively. e is the edge connecting the two vertices. Its length is ∣e∣.
Fig 5.
Cell geometry is determined by the ratio of the tension coefficients. a) when η(i,j) = 0, there is no tension on an inner edge, and it can be regarded as an imaginary cell wall; b) when η(i,j) = 0.5η(i,0), there is a strong adhesion force between the two cells; c) when η(i,j) = η(i,0) = η(j,0), the two cells behave as if physically they have the same wall; d) when η(i,j) = 1.5η(i,0), there is a weak adhesion force between the two cells; e) when η(i,j) ≥ 2η(i,0), The two cells have no adhesion and behave like soccer balls. Adding an inner wall would be more costly, as it is equivalent to adding two outer walls. In this case, the overall energy of the two cells is not reduced.
Table 2.
Algorithm 1. UpdateCellPattern (V(t), Δ (t), Δη(t), σ, k).
Fig 6.
Possible topological changes when cells grow or when their boundaries move. (a) Edge insertion. When two isolated cells come into contact, we add an edge to represent the intersection plane of the boundary between these two cells. (b) Void removal. When three cells come into contact, the curve triangular empty space is replaced by a vertex where the inner edges meet. (c) Edge flip. When two previously disconnected cells expand to meet while pushing away two previously connected cells, we replace one inner edge with a new inner edge.
Fig 7.
Visualization of the growth of tissue from 2 to 4,000 cells.
a) 2 cells; b) 100 cells; c) 500 cells; d) 1,000 cells; e) 4,000 cells, and f) zoomed in view of 4,000 cells.
Fig 8.
Model of tissue development starting from a single cell.
(a) A single cell and its plane of first division; (b) Two daughter cells after the first division, each is slight deformed from its shape in (a); (c) The formation of four cells after two cell divisions.
Fig 9.
(a) and (b) shows two separate growing cells come into contact and become fused together. A new edge is formed between the two cells. (c) and (d) shows the case of three growing cells fusion. Three new edges and a new vertex are formed after fusion. (e) and (f) shows the fusion of two growing tissues. Two separate tissues contact with each other and become fused together to form a continuous tissue. The new edges and vertices formed are highlighted.
Fig 10.
Apoptosis of a peripheral cell.
(a) A small tissue before the labeled red cell proceeds to apoptosis. (b) The tissue after the demise of the red cell. The black arrow points to the location where the red cell was. The apoptosis of the red cell caused significant changes on its neighboring cells (labeled 1 and 2). A larger tissue before (c) and after (d) the labeled red cell proceeds to apoptosis. The black arrow in (d) points to the location where the red cell was apoptosized.
Fig 11.
Cartoon representation of the bristles on the epidermis of fruit fly. Most bristles are organized in regular rows that are parallel or perpendicular to the body axis or limbs. The bristles are evenly spaced and aligned within each of the rows. Modified from http://www.biology-resources.com/drawing-fruit-fly.html
Fig 12.
The “Pre-destined Model” suggested that bristle sites can form based on patterns of expression of genes (ac and sc genes) [80]. Here sharp expression boundaries indicate bristle sites. Our simulation results show that the “Pre-destined Model” can lead to aggregation of bristles, which are not observed experimentally. For example, red circles highlight instances where ≥ 2 contacting bristle cells form aggregates.
Fig 13.
Simulation results of bristle pattern formation using different models. a) The pattern of gene expression used for the stripe models. Green stripes have almost equal expression of Dl and N genes but ac is not expressed. Blue stripes have high expression of ac and Dl genes but low expression of N gene. Red stripes have high expression of ac and N genes but low expression of Dl gene. Bristles only form in the blue stripes. b) Lateral inhibition with stripes does not ensure equal spacing or good alignment. c) Inhibition field with out stripes ensures proper spacing but does not produce a good alignment. d) Inhibition field with stripes produces equal spacing as well as good alignment.
Fig 14.
Steady state spatial gradients of Dl of different diffusion rates.
The steady state spatial gradients of Dl are due to diffusion and degradation. The red solid line with triangular markers is the steady state gradient formed with the diffusion coefficient 1.2 μm2 s−1. The green dash line with square markers is the gradient of the diffusion coefficient 4.8 μm2 s−1. The blue dotted line with circle markers is of the diffusion coefficient 9.6 μm2 s−1. The black straight lines represents the 0.05 Dl concentration threshold for cellular response.
Fig 15.
Relationship between stripe width and Inhibition Radius.
Effects of stripe width and inhibition radius on the alignment of the bristles. (Top) The stripe width increase from left to right and the inhibition field radius increases from top to bottom. For a fixed Inhibition radius (horizontal rows) small stripe width produces a better alignment (Alignment Index ϱ is smaller). For a fixed stripe width (horizontal rows), large inhibition field radius produces a better alignment. (Bottom) The degree of alignment of bristle cells increases as the stripe width is reduced and inhibition radius is increased, suggesting that the alignment is directly proportional to the inhibition radius and inversely proportional to the stripe width.
Fig 16.
The degree of alignment by an index ϱ.
It is calculated by drawing a vertical line in the middle of each stripe and summing the number of cells that lie between the bristles and the line. This is then normalized with the total number of bristles in the stripe. Here, a) is an example of good alignment with ϱ = 0.5 and b) is an example of bad alignment with ϱ = 2.0
Table 3.
Simulation results for “Pre-destined”, “Lateral inhibition”, “Lateral inhibition with stripes”, “Inhibition Field”, and “Inhibition Field with Stripes” models.
Fig 17.
Model of feedback circuits for tissue size control.
a) Division types of stem cells and progenitor cells. Red sphere labeled with (S) indicates stem cells, blue hexagon (P) indicates progenitor cells, and white diamond (D) indicates differentiated cell. The same color code is used for illustration of resulting tissues. b) Feedback controls of stem cell model. Blue arrows indicate self-renewal or proliferation divisions. Black arrows indicate symmetric differentiation divisions. Red arrows indicate asymmetric divisions. Flat-head arrows extending from differentiated cell with corresponding colors indicate inhibitions to respective type of divisions.
Fig 18.
Effects of different inhibition ranges of secreted factors in negative feedback loop to stem cells on tissue size control.
Examples of tissue pattern and the time course of population size of different cell types when stem cells are inhibited by differentiated cells located within (a,d) 3-, (b,e) 2-, and (c,f) 4-layers of neighboring cells, respectively. Normal size control with the ability to regenerate is achieved when 3 layers of neighboring cells are inhibited (a, and d). When 2 layers of cells are inhibited, size control is no longer possible (b and e). When 4 layers of cells are inhibited, tissue size is suppressed.
Fig 19.
Effects of inhibitions to stem cell symmetric and asymmetric divisions.
An example of tissue formation without inhibition to symmetric division is shown in a), and its corresponding time-course of the size of population of different cell types is shown in b). c) An example of tissue formation without inhibition to asymmetric division and d) the corresponding time course of size of populations of different cell types.
Fig 20.
Butterfly Scales with Inhibition Field radius of 2 cells.
Simulation with Inhibition field radius of 2 cells produces evenly spaced cells but regular rows do not form.